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How do I solve the following equation:

$$x + y\ne0\text{ and }\frac{1}{x+y}=x$$

Wolfram Alpha came up with this solution

$$x\ne0,\:y=\frac{1-x^2}{x}$$

but I don't know how to get there.

thx alex

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1 Answer

up vote 1 down vote accepted

First af all, $x + y \neq 0$ since it is a denominator. Therefore $1=x(x+y)$. If $x=0$, then $1=0$, a contradiction. Therefore $x\neq 0$, and we can divide by $x$: $$\frac{1}{x}=x+y.$$ Now $$y=\frac{1}{x}-x = \frac{1-x^2}{x},$$ under the condition that $x \neq 0$.

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